In applications such as multi-receiver radars and ultrasound array systems, the observed signals are often modeled as the convolution of the transmit pulse signal and a set of sparse filters representing the sparse target scenes. A sparse multichannel blind deconvolution (MBD) problem simultaneously identifies the unknown signal and sparse filters, which is in general ill-posed. In this paper, we consider the identifiability problem of sparse-MBD and show that, similar to compressive sensing, it is possible to identify the sparse filters from compressive measurements of the output sequences. Specifically, we consider compressible measurements in the Fourier domain and derive deterministic identifiability conditions. Our main results demonstrate that L-sparse filters can be identified from 2L2 Fourier measurements from two or more coprime channels. We also show that 2L measurements per channel are necessary. The sufficient condition sharpens as the number of channels increases and is asymptotically optimal, i.e., it suffices to acquire on the order of L Fourier samples per channel. We also propose a kernel-based sampling scheme that acquires Fourier measurements from a commensurate number of time-domain samples. The gap between the sufficient and necessary conditions is illustrated through numerical experiments including comparing practical reconstruction algorithms. The proposed compressive MBD results require fewer measurements and fewer channels for identifiability compared to previous results, which aids in building cost-effective receivers.

中文翻译：

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压缩多通道盲解卷积的可识别性条件


在多接收器雷达和超声阵列系统等应用中，观测到的信号通常被建模为发射脉冲信号和表示稀疏目标场景的一组稀疏滤波器的卷积。稀疏多通道盲反卷积（MBD）问题同时识别未知信号和稀疏滤波器，这通常是不适定的。在本文中，我们考虑了稀疏MBD的可识别性问题，并表明，与压缩感知类似，可以从输出序列的压缩测量中识别稀疏滤波器。具体来说，我们考虑傅里叶域中的可压缩测量并推导确定性可识别性条件。我们的主要结果表明，可以通过两个或多个互质通道的 2L2 傅立叶测量来识别 L 稀疏滤波器。我们还表明每个通道 2L 测量是必要的。充分条件随着通道数量的增加而变得尖锐，并且是渐近最优的，即，足以在每个通道上采集L个傅里叶样本。我们还提出了一种基于内核的采样方案，该方案从相应数量的时域样本中获取傅立叶测量值。通过数值实验（包括比较实际的重建算法）来说明充分条件和必要条件之间的差距。与之前的结果相比，所提出的压缩 MBD 结果需要更少的测量和更少的通道来实现可识别性，这有助于构建具有成本效益的接收器。